Vibration of plates

Remarks

Example: a simply supported rectangle $0 < x < a$, $0 < y < b$ has the modes $W_{mn} = \sin(m\pi x/a) \sin(n\pi y/b)$ for integers $m, n \ge 1$. Check: $\Delta^2 W_{mn} = \pi^4 (m^2/a^2 + n^2/b^2)^2 W_{mn}$, and $W_{mn} = \Delta W_{mn} = 0$ on the edges, so $$\omega_{mn} = \pi^2 \left( \frac{m^2}{a^2} + \frac{n^2}{b^2} \right) \sqrt{\frac{D}{\rho h}}.$$ Its nodal lines are the segments $x = ka/m$ with $0 < k < m$ and $y = lb/n$ with $0 < l < n$.

Non-example: $W = \sin(\pi x/a)$ on the same rectangle solves $D \Delta^2 W = \rho h \omega^2 W$ for a suitable $\omega$. It does not vanish on the edges $y = 0$ and $y = b$, so it is not a mode of the simply supported plate.

For a bounded plate with clamped or simply supported edges the natural frequencies form an increasing sequence tending to infinity. The mode shapes are orthogonal in $L^2(\Omega)$. This holds because $\omega^2$ are the eigenvalues of a positive self-adjoint operator with compact resolvent.